U.S. patent application number 11/683195 was filed with the patent office on 2007-09-27 for method and apparatus for transforming overbounds.
This patent application is currently assigned to EADS Astrium GmbH. Invention is credited to Harald Frankenberger, Hans L. Trautenberg.
Application Number | 20070222669 11/683195 |
Document ID | / |
Family ID | 36940313 |
Filed Date | 2007-09-27 |
United States Patent
Application |
20070222669 |
Kind Code |
A1 |
Trautenberg; Hans L. ; et
al. |
September 27, 2007 |
METHOD AND APPARATUS FOR TRANSFORMING OVERBOUNDS
Abstract
A method for determining overbounds comprises the steps of
determining conservative overbounds (q.sub.i) of at least one error
(.epsilon..sub.i) in a first phase space, multiplying the
conservative overbounds (q.sub.i) of errors (.epsilon..sub.i) in
the first phase space by a first parameter (.theta.(-x)2) and a
second parameter (.theta.(x)2), and determining an upper bound for
the integrity risk at the alert limit (p.sub.w,int(AL)) in a second
phase space using overbounds (q.sub.i) of errors (.epsilon..sub.i)
in the first phase space by the first parameter (.theta.(-x)2) and
the second parameter (.theta.(x)2).
Inventors: |
Trautenberg; Hans L.;
(Ottobrunn, DE) ; Frankenberger; Harald;
(Ottobrunn, DE) |
Correspondence
Address: |
CROWELL & MORING LLP;INTELLECTUAL PROPERTY GROUP
P.O. BOX 14300
WASHINGTON
DC
20044-4300
US
|
Assignee: |
EADS Astrium GmbH
Muenchen
DE
|
Family ID: |
36940313 |
Appl. No.: |
11/683195 |
Filed: |
March 7, 2007 |
Current U.S.
Class: |
342/127 ;
342/357.395; 342/357.45; 342/357.58 |
Current CPC
Class: |
G01S 19/08 20130101;
G01S 19/20 20130101 |
Class at
Publication: |
342/127 |
International
Class: |
G01S 13/08 20060101
G01S013/08 |
Foreign Application Data
Date |
Code |
Application Number |
Mar 8, 2006 |
DE |
06 004 754.5 |
Claims
1. A method for determining limiting overbounds for distributions
generated by a system, said method comprising: determining
overbounds (q.sub.i) of at least one error (.epsilon..sub.i) in a
first phase space; multiplying the overbounds (q.sub.i) of errors
(.epsilon..sub.i) in the first phase space by a first parameter
(.theta.(-x)2) and a second parameter (.theta.(x)2); and
determining an upper bound for the integrity risk at an alert limit
(P.sub.w,int(AL)) in a second phase space using overbounds
(q.sub.i) of errors (.epsilon..sub.i) in the first phase space by
the first parameter (.theta.(-x)2) and the second parameter
(.theta.(x)2).
2. The method according to claim 1, where the first phase space is
the range domain and the second phase space is the position
domain.
3. The method according to claim 1, where the first parameter
(.theta.(-x)2) and the second parameter (.theta.(x)2) are equal and
constant.
4. A computer readable medium encoded with a program for
determining limiting overbounds for distributions generated by a
system, by performing the following steps: determining overbounds
(q.sub.i) of at least one error (.epsilon..sub.i) in a first phase
space; multiplying the overbounds (q.sub.i) of errors
(.epsilon..sub.i) in the first phase space by a first parameter
(.theta.(-x)2) and a second parameter (.theta.(x)2); and
determining an upper bound for the integrity risk at an alert limit
(p.sub.w,int(AL)) in a second phase space using overbounds
(q.sub.i) of errors (.epsilon..sub.i) in the first phase space by
the first parameter (.theta.(-x)2) and the second parameter
(.theta.(x)2).
5. The computer readable medium according to claim 4, where the
first phase space is the range domain and the second phase space is
the position domain.
6. The computer readable medium according to claim 4, where the
first parameter (.theta.(-x)2) and the second parameter
(.theta.(x)2) are equal and constant.
Description
BACKGROUND OF THE INVENTION
[0001] This application claims the priority of European patent
document 06 004 754.5, filed Mar. 8, 2006, the disclosure of which
is expressly incorporated by reference herein.
[0002] The invention relates to a method and apparatus for
transferring a Galileo overbound or an ICAO overbound to a pairwise
overbound with excess mass (POEM).
[0003] For Global Navigation Satellite Systems (GNSS) based
navigation systems for aviation, it must be assured that the
position the system provides has sufficient integrity. This means
that the probability that the navigation system supplies
hazardously misleading information (HMI) should be proven to remain
extremely small under all circumstances. The problem of trying to
guarantee that such a system offers sufficient integrity is known
as the overbounding problem, because practical solutions are
necessarily conservative (bounding) with respect to the performance
that is actually obtained. Further, Safety-of-life (SoL) GNSS
augmentation systems must provide bounds on the probability that
hazardous navigation errors may occur.
[0004] The integrity information sent to the user contains no
explicit provisions for protecting against biases. Instead users
are sent protection factors that correspond to zero-mean error
distributions. The users combine the received protection factors
using their own local knowledge to calculate protection levels that
correspond to their position estimate. The broadcast protection
factors must be sufficient such that any individual user has only a
small probability (e.g. less than a one in ten million), for each
approach, that their true position error exceeds the calculated
protection level. The ground system for instance must guarantee
these protection factors without knowing precisely where the users
are, or which satellite they observe.
[0005] For determining the system's integrity, errors in the range
domain are transformed into errors in the position domain. During
the transformation of the errors in the range domain into the
errors in the position domain, the corresponding error statistics
(probability distribution functions of the errors) are transformed
by the convolution which is necessary for such transformation.
[0006] In the literature, several different overbounding concepts
are known. One of these concepts is the Galileo overbounding as
defined by the Galileo requirements. Another concept is the paired
overbounding with excess mass (POEM). The Galileo overbounding has
the disadvantage that it is not preserved during convolution,
whereas the POEM is preserved during convolution. That means that
the convolution of two excess mass overbounding distributions will
overbound the convolution of the two original convolutions. Thus,
the paired overbounding concept effectively relates range domain
and position domain overbounding.
SUMMARY OF THE INVENTION
[0007] One object of the present invention is to define a process
and an apparatus that transforms an overbound that is not preserved
during convolution into an overbound that is preserved during
convolution.
[0008] This and other objects and advantages are achieved by the
method and apparatus according to the invention, in which the
Galileo Overbounding definition of a distribution is used to define
parameters for a paired overbounding with excess mass (POEM) of the
same distribution. This is necessary as the properties of the
Galileo overbounding definition are not preserved during
convolutions of distributions, whereas the paired overbounding with
excess mass properties are preserved during convolutions of
distributions. The convolution is necessary for the transformation
from the range domain to the position domain.
DETAILED DESCRIPTION OF THE INVENTION
[0009] Before describing several embodiments of the invention
several overbounding definitions are stated.
Galileo Overbounding Definition
[0010] For Galileo the probability density p is overbounded by a
function q if the equation .intg. - .infin. - y .times. p
.function. ( x ) .times. d x + .intg. y .infin. .times. p
.function. ( x ) .times. d x .ltoreq. .intg. - .infin. - y .times.
q .function. ( x ) .times. d y + .intg. y .infin. .times. q
.function. ( x ) .times. d x .times. .times. .times. for .times.
.times. all .times. .times. y .gtoreq. 0 ( 0.1 ) ##EQU1## holds
true.
[0011] It has to be noted further, that for Galileo it is foreseen
to use as the overbounding distributions only Gaussian
distributions of the form q .function. ( t ) .ident. 1 2 .times.
.pi. .times. .sigma. .times. e - t 2 2 .times. .sigma. 2 . ( 0.2 )
##EQU2##
[0012] It is worthwhile to note, that q is symmetric zero mean
q(t)=q(-t), (0.3) zero mean .intg. - .infin. .infin. .times. t q
.function. ( t ) .times. d t = 0 , ( 0.4 ) .intg. - .infin. 0
.times. q .function. ( t ) .times. d t = .intg. 0 .infin. .times. q
.function. ( t ) .times. d t = 0.5 , ( 0.5 ) ##EQU3## and that
.intg. a b .times. q .function. ( x ) .gtoreq. 0 .times. .times.
for .times. .times. any .times. .times. b .gtoreq. a . ( 0.6 )
##EQU4##
[0013] It is worthwhile to note that .intg. a b .times. p
.function. ( x ) .gtoreq. 0 .times. .times. for .times. .times. any
.times. .times. b .gtoreq. a ( 0.7 ) ##EQU5## and that .intg. a b
.times. p .function. ( x ) .ltoreq. 1 .times. .times. for .times.
.times. any .times. .times. b , a ( 0.8 ) ##EQU6##
[0014] It is known from literature, that the property defined by
equation (0.1) is not preserved during convolution of
distributions.
Paired Overbounding with Excess Mass Definition
[0015] A probability density p is paired overbounded by the
functions q.sub.L and q.sub.R, if the equation .intg. - .infin. y
.times. q L .function. ( x ) .times. d x .gtoreq. .intg. - .infin.
y .times. p .function. ( x ) .times. d x .gtoreq. 1 - .intg. y
.infin. .times. q R .function. ( x ) .times. d x .times. .times.
for .times. .times. all .times. .times. y ( 0.9 ) ##EQU7## holds
true. The functions q.sub.L/R have to fulfil the following
requirements. q L / R .function. ( x ) .gtoreq. 0 .times. .times.
for .times. .times. all .times. .times. x .times. .times. and (
0.10 ) .intg. - .infin. .infin. .times. q L / R .function. ( x )
.times. d x = K L / R .gtoreq. 1 ( 0.11 ) ##EQU8##
[0016] It is known that the property defined by equation (0.9) is
preserved during convolution and scaling. To ensure that the
convolutions can be performed analytically it is convenient to
define q.sub.L and q.sub.R as follows: q L .function. ( x ) = K L 2
.times. .pi. .times. .sigma. L .times. e - ( x - b L ) 2 2 .times.
.sigma. L 2 ( 0.12 ) q R .function. ( x ) = K R 2 .times. .pi.
.times. .sigma. R .times. e - ( x - b R ) 2 2 .times. .sigma. R 2 (
0.13 ) ##EQU9##
[0017] If the individual contributions of the range errors
.epsilon..sub.i are paired overbounded with excess mass by the
functions q L , i .function. ( x ) = K L , i 2 .times. .times. .pi.
.times. .sigma. L , i .times. e - ( x - b L , i ) 2 2 .times.
.sigma. L , i 2 .times. .times. and ( 0.14 ) q R , i .function. ( x
) = K R , i 2 .times. .pi. .times. .sigma. R , i .times. e - ( x -
b R , i ) 2 2 .times. .sigma. R , i 2 ( 0.15 ) ##EQU10## and if the
errors in the range domain .epsilon..sub.i are mapped onto the
error in the position domain .epsilon..sub.pos by pos = i = 1 n
.times. M w , i i ( 0.16 ) ##EQU11## an upper bound for the
integrity risk at the alert limit p.sub.w,int (AL) in the direction
w is given by p w , int .function. ( AL ) .ltoreq. K L , M w + K R
, M w 2 - K R , M w 2 .times. .times. erf .times. .times. ( AL - b
R , M w 2 .times. .sigma. R , M w ) + K L , M .times. w 2 .times.
.times. erf .times. .times. ( - AL - b L , M w 2 .times. .sigma. L
, M w ) .times. .times. with ( 0.17 ) g .function. ( .alpha. ) = {
R , if .times. .times. .alpha. > 0 L , if .times. .times.
.alpha. < 0 ( 0.18 ) k .function. ( .alpha. ) = { L , if .times.
.times. .alpha. > 0 R , if .times. .times. .alpha. < 0 ( 0.19
) K R , M w = i = 1 n .times. K g .function. ( M w , i ) , i ( 0.20
) K L , M w = i = 1 n .times. K k .function. ( M w , i ) , i ( 0.21
) b R , M w = i = 1 n .times. M w , i .times. b g .function. ( M w
, i ) , i ( 0.22 ) b L , M w = i = 1 n .times. M w , i .times. b k
.function. ( M w , i ) , i ( 0.23 ) .sigma. R , M w = i = 1 n
.times. ( M w , i .times. .sigma. g .function. ( M w , i ) , i ) 2
( 0.24 ) .sigma. L , M w = i = 1 n .times. ( M w , i .times.
.sigma. k .function. ( M w , i ) , i ) 2 ( 0.25 ) ##EQU12## ICAO
Overbounding
[0018] For ICAO the probability density p is overbounded by a
function q if the equations .intg. - .infin. - y .times. p
.function. ( x ) .times. d x .ltoreq. .intg. - .infin. - y .times.
q .function. ( x ) .times. d y .times. .times. for .times. .times.
all .times. .times. y .gtoreq. 0 ( 0.26 ) .intg. y .infin. .times.
p .function. ( x ) .times. d x .ltoreq. .intg. y .infin. .times. q
.function. ( x ) .times. d y .times. .times. for .times. .times.
all .times. .times. y .gtoreq. 0 ( 0.27 ) ##EQU13## hold true.
[0019] The ICAO overbounding definition implies directly the
Galileo overbounding definition. This can be seen by a simple
addition of the defining inequalities. The opposite is not valid in
general.
[0020] It has to be noted further, that for ICAO it is foreseen to
use as the overbounding distributions only Gaussian distributions
of the form q .function. ( t ) .ident. 1 2 .times. .pi. .times.
.sigma. .times. e - t 2 2 .times. .sigma. 2 . ( 0.28 )
##EQU14##
[0021] It has to be further noted that ICAO states that for the
receiver contribution to the error it can be assumed that
p(x)=p(-x), (0.29) p(sign(x)(|x|+.epsilon.)).ltoreq.p(x) for all x
and for all .epsilon..gtoreq.0 (0.30) that is probability density p
is symmetric and monotonically increasing up to a single mode in
x=0 and then monotonically decreasing (p is also called unimodal).
Process for Mapping of Galileo Overbounding to Poem
[0022] Mapping without bias. As a first embodiment the process for
mapping of Galileo overbounding to POEM for the case without bias
is described.
[0023] For a mapping of the Galileo overbounding to POEM without
bias we define q.sub.L/R.ident.2q (0.31)
[0024] We then compute, observing that q is symmetric, .intg. -
.infin. y .times. q L .function. ( x ) .times. d x = .intg. -
.infin. y .times. 2 .times. q .function. ( x ) .times. d x = .intg.
- .infin. y .times. q .function. ( x ) .times. d x + .intg. - y
.infin. .times. q .function. ( x ) .times. d x . ( 0.32 )
##EQU15##
[0025] For y.ltoreq.0 it follows now from (0.32), (0.1), and (0.7)
that .intg. - .infin. y .times. q L .function. ( x ) .times. d x
.gtoreq. .intg. - .infin. y .times. p .function. ( x ) .times. d y
+ .intg. - y .infin. .times. p .function. ( x ) .times. d x
.gtoreq. .intg. - .infin. y .times. p .function. ( x ) .times. d y
. ( 0.33 ) ##EQU16##
[0026] For y.gtoreq.0 it follows now from (0.32), the first part of
(0.5), and (0.8) that .intg. - .infin. y .times. q L .function. ( x
) .times. d x = .intg. - .infin. 0 .times. q L .function. ( x )
.times. d x + .intg. 0 y .times. q L .function. ( x ) .times. d x =
1 + .intg. 0 y .times. q L .function. ( x ) .times. d x .gtoreq. 1
.gtoreq. .intg. - .infin. y .times. p .function. ( x ) .times. d y
. ( 0.34 ) ##EQU17##
[0027] So it has been shown that with the definition given in
(0.31) and provided that (0.1) hold true the following condition
holds true for all y .intg. - .infin. y .times. q L .function. ( x
) .times. d x .gtoreq. .intg. - .infin. y .times. p .function. ( x
) .times. d y ( 0.35 ) ##EQU18##
[0028] We now compute, observing that q is symmetric, 1 - .intg. y
.infin. .times. q R .function. ( x ) .times. d x = 1 - .intg. y
.infin. .times. 2 .times. q .function. ( x ) .times. d x = 1 -
.intg. - .infin. - y .times. q .function. ( x ) .times. d x -
.intg. y .infin. .times. q .function. ( x ) .times. d x ( 0.36 )
##EQU19##
[0029] For y.gtoreq.0it follows from (0.36), (0.1), and (0.7) 1 -
.intg. y .infin. .times. q R .function. ( x ) .times. d x = 1 -
.intg. - .infin. - y .times. q .function. ( x ) .times. d x -
.intg. y .infin. .times. q .function. ( x ) .times. d x .ltoreq. 1
- .intg. - .infin. - y .times. p .function. ( x ) .times. d x -
.intg. y .infin. .times. p .function. ( x ) .times. d x .ltoreq. 1
- .intg. y .infin. .times. p .function. ( x ) .times. d x ( 0.37 )
##EQU20##
[0030] For y.ltoreq.0 if follows from the first part of (0.36),
(0.5), (0.6), and (0.8) 1 - .intg. y .infin. .times. q R .function.
( x ) .times. d x = 1 - .intg. 0 .infin. .times. 2 .times. q
.function. ( x ) .times. d x - .intg. y 0 .times. 2 .times. q
.function. ( x ) .times. d x = - .intg. y 0 .times. 2 .times. q
.function. ( x ) .times. d x .ltoreq. 0 .ltoreq. 1 - .intg. y
.infin. .times. p .function. ( x ) .times. d x ( 0.38 )
##EQU21##
[0031] So it has been shown with (0.37) and (0.38) that for any y
the following holds true: 1 - .intg. y .infin. .times. q R
.function. ( x ) .times. d x .ltoreq. 1 - .intg. y .infin. .times.
p .function. ( x ) .times. d x = .intg. - .infin. y .times. p
.function. ( x ) .times. d x ( 0.39 ) ##EQU22##
[0032] Combing (0.35) and (0.39) we finally get .intg. - .infin. y
.times. q L .function. ( x ) .times. d x .gtoreq. .intg. - .infin.
y .times. p .function. ( x ) .times. d y .gtoreq. 1 - .intg. y
.infin. .times. q R .function. ( x ) .times. d x ( 0.40 ) ##EQU23##
for any y.
[0033] Assuming now m satellites and on each range two different
types of contributions of range errors, says .epsilon..sub.i and
.epsilon..sub.m+i on range i.epsilon.{1, . . . ,m}, e.g. one type
due to the system in space (errors from orbit, satellite, clock,
etc.) and one due to the local effects at the receiver location
(errors from atmosphere, receiver noise, multipath, etc.). Let
n=2m. Then using definition (0.31 ) results in
K.sub.L,i=K.sub.R,i=2 for all i=1, . . . ,n (0.41)
b.sub.L,i=b.sub.R,i=0 for all i=1, . . . ,n (0.42)
.sigma..sub.L,i=.sigma..sub.R,i=.sigma..sub.i=SISA.sub.i for all
i=1, . . . ,m and
.sigma..sub.L,i=.sigma..sub.R,i=.sigma..sub.i=.sigma..sub.u,i-m for
all i=m+1, . . . ,n (0.43) for the fault free case.
[0034] Taking (0.41), (0.42), (0.43) and the equality
-erf(x)=erf(-x) into account, (0.17) simplifies to p w , int
.function. ( AL ) .ltoreq. 2 n .times. ( 1 - erf ( AL 2 .times. i =
1 n .times. ( M w , i .times. .sigma. i ) 2 ) ) ( 0.44 )
##EQU24##
[0035] Observing M.sub.i=M.sub.m+i for all i=1, . . . , m because
these factors depend only on the geometry given by the m satellites
and the receiver location but not on the special type of error
contribution, this formular can be written as p w , int .function.
( AL ) .ltoreq. 4 m .times. ( 1 - erf ( AL 2 .times. i = 1 m
.times. M w , i 2 .function. ( SISA i 2 + .sigma. u , i 2 ) ) ) (
0.45 ) ##EQU25##
[0036] For the faulty case, satellite j.epsilon.{1, . . . , m} is
faulty, we get K L , i = K R , i = 2 .times. .times. for .times.
.times. all .times. .times. i = 1 , .times. , n ( 0.46 ) b L , i =
b R , i = 0 .times. .times. for .times. .times. all .times. .times.
i = 1 , .times. , n .times. .times. with .times. .times. i .noteq.
j .times. .times. and .times. - b L , j = b R , j = TH j ( 0.47 )
.sigma. L , i = .sigma. R , i = .sigma. i = SISA i .times. .times.
for .times. .times. all .times. .times. i = 1 , .times. , m .times.
.times. with .times. .times. i .noteq. j .times. .times. and
.times. .times. .sigma. L , i = .sigma. R , i = .sigma. i = .sigma.
u , i - m .times. .times. for .times. .times. all .times. .times. l
= m + 1 , .times. , n .times. .times. and .times. .times. .sigma. L
, j = .sigma. R , j = .sigma. j = SISMA j ( 0.47 ' ) p w , int
.function. ( AL ) .ltoreq. 2 n .times. ( 1 - erf ( AL - M w , j
.times. TH j 2 .times. i = 1 n .times. ( M w , i .times. .sigma. i
) 2 ) ) ( 0.48 ) ##EQU26##
[0037] Again observing M.sub.i=M.sub.m+i for all i=1, . . . , m
this formular can be written as p w , int .function. ( AL )
.ltoreq. 4 m .times. ( 1 - erf ( AL - M w , j .times. TH j 2
.times. i = 1 , i .noteq. j m .times. M w , i 2 .times. ( SISA i 2
+ .sigma. u , i 2 ) + 2 .times. M w , j 2 .function. ( SISMA j 2 +
.sigma. u , j 2 ) ) ) ( 0.49 ) ##EQU27##
[0038] Mapping with bias. As a second embodiment the process for
mapping of Galileo overbounding to POEM for the case with bias is
described.
[0039] For a mapping of the Galileo overbounding to POEM with bias
we define q'.sub.L(x).ident..theta.(-x)2q(x) (0.50)
q'.sub.R(x).ident..theta.(x)2q(x) (0.51) where .theta. is a
function, defined for real values x by .theta. .function. ( x ) = {
1 , if .times. .times. x .gtoreq. 0 0 , otherwise . ##EQU28##
[0040] For y.ltoreq.0 we then compute, observing that q is
symmetric, .intg. - .infin. y .times. q L ' .function. ( x )
.times. d x = .intg. - .infin. y .times. 2 .times. q .function. ( x
) .times. d x = .intg. - .infin. - y .times. q L ' .function. ( x )
.times. d x + .intg. - y .infin. .times. q .function. ( x ) .times.
d x . ( 0.52 ) ##EQU29##
[0041] It follows now from (0.52), (0.1), and (0.7) that .intg. -
.infin. y .times. q L ' .function. ( x ) .times. d x .gtoreq.
.intg. - .infin. y .times. p .function. ( x ) .times. d y + .intg.
- y .infin. .times. p .function. ( x ) .times. d x .gtoreq. .intg.
- .infin. y .times. p .function. ( x ) .times. d y . ( 0.53 )
##EQU30##
[0042] For y.gtoreq.0 it follows from first part of (0.5), and
(0.8) that .intg. - .infin. y .times. q ' L .function. ( x )
.times. .times. d x = .times. .intg. - .infin. 0 .times. q ' L
.function. ( x ) .times. .times. d x + .intg. 0 y .times. q ' L
.function. ( x ) .times. .times. d x = .times. 1 + 0 .gtoreq. 1
.gtoreq. .intg. - .infin. y .times. p .function. ( x ) .times.
.times. d y . ( 0.54 ) ##EQU31##
[0043] So it has been shown that with the definition given in
(0.50) and provided that (0.1) hold true the following condition
holds true for all y .intg. - .infin. y .times. q ' L .function. (
x ) .times. .times. d x .gtoreq. .intg. - .infin. y .times. p
.function. ( x ) .times. .times. d y ( 0.55 ) ##EQU32##
[0044] We now compute for y.gtoreq.0 and observing that q is
symmetric, 1 - .intg. y .infin. .times. q ' R .function. ( x )
.times. .times. d x = 1 - .intg. y .infin. .times. 2 .times.
.times. q .times. ( x ) .times. .times. d x = 1 - .intg. - .infin.
- y .times. q .function. ( x ) .times. .times. d x - .intg. y
.infin. .times. q .function. ( x ) .times. .times. d x ( 0.56 )
##EQU33##
[0045] It follows from (0.56), (0.1), and (0.7) 1 - .intg. y
.infin. .times. q ' R .function. ( x ) .times. .times. d x =
.times. .intg. - .infin. - y .times. q .function. ( x ) .times.
.times. d x - .intg. y .infin. .times. q .function. ( x ) .times.
.times. d x .ltoreq. .times. 1 - .intg. - .infin. - y .times. p
.function. ( x ) .times. .times. d x - .intg. y .infin. .times. p
.function. ( x ) .times. .times. d x .ltoreq. .times. 1 - .intg. y
.infin. .times. p .function. ( x ) .times. .times. d x ( 0.57 )
##EQU34##
[0046] For y.ltoreq.0 if follows from (0.5), (0.6), and (0.8) 1 -
.intg. y .infin. .times. q ' R .function. ( x ) .times. .times. d x
= .times. 1 - .intg. 0 .infin. .times. q ' R .function. ( x )
.times. .times. d x = 1 - .intg. 0 .infin. .times. 2 .times.
.times. q .function. ( x ) .times. .times. d x = .times. 0 .ltoreq.
1 - .intg. y .infin. .times. p .function. ( x ) .times. .times. d x
( 0.58 ) ##EQU35##
[0047] So we have shown with (0.57) and (0.58) that for any y the
following holds true: 1 - .intg. y .infin. .times. q ' R .function.
( x ) .times. .times. d x .ltoreq. 1 - .intg. y .infin. .times. p
.function. ( x ) .times. .times. d x = .intg. - .infin. y .times. p
.function. ( x ) .times. .times. d x ( 0.59 ) ##EQU36##
[0048] Combining (0.55) and (0.59) we finally get .intg. - .infin.
y .times. q ' L .function. ( x ) .times. .times. d x .gtoreq.
.intg. - .infin. y .times. p .function. ( x ) .times. .times. d y
.gtoreq. 1 - .intg. y .infin. .times. q ' R .function. ( x )
.times. .times. d x ( 0.60 ) ##EQU37## for any y.
[0049] Assuming again as above m satellites and on each range two
different types of contributions of range errors, say and
.epsilon..sub.i and .epsilon..sub.m+i on range i.epsilon.{1, . . .
, m}, e.g. one due to the system in space (errors from orbit,
satellite, clock, etc.) and one due to the local effects at the
receiver location (errors from atmosphere, receiver noise,
multipath, etc.). Let n=2m. Then it is possible to prove that
probability densities p.sub.i are paired overbounded by the
functions q.sub.L,i and q.sub.R,i defined by equations (0.14) and
(0.15) with .sigma. L , i = .sigma. R , i = .sigma. i = SISA i
.times. .times. for .times. .times. all .times. .times. i = 1 ,
.times. , m .times. .times. and .times. .times. .times. .times.
.sigma. L , i = .sigma. R , i = .sigma. i = .sigma. u , i - m
.times. .times. for .times. .times. all .times. .times. i = m + 1 ,
.times. , n ( 0.61 ) - b L , i = b R , i = b i > 0 .times.
.times. for .times. .times. all .times. .times. i = 1 , .times. , n
( 0.62 ) K L , i = K R , i = K i = 2 ( 1 + erf ( b i .sigma. i
.times. 2 ) ) - 1 .times. .times. for .times. .times. all .times.
.times. i = 1 , .times. , n ( 0.63 ) ##EQU38## for the fault free
case.
[0050] Taking these definitions into account and remembering
equality -erf(x)=erf(-x), equation (0.17) simplifies to p w , int
.function. ( AL ) .ltoreq. 2 n i = 1 n .times. .times. ( 1 + erf (
b i .sigma. i .times. 2 ) ) ( 1 - erf .function. ( AL - i = 1 n
.times. .times. M w , i .times. b i 2 .times. i = 1 n .times.
.times. ( M w , i .times. .sigma. i ) 2 ) ) ( 0.64 ) ##EQU39##
[0051] Observing again M.sub.i=M.sub.m+i for all i=1, . . . m
because these factors depend only on the geometry given by the m
satellites and the receiver location but not on the special type of
error contribution, this formular can be written as p w , int
.function. ( AL ) .ltoreq. 4 m i = 1 m .times. .times. ( 1 + erf (
b i SISA i .times. 2 ) ) i = 1 m .times. .times. ( 1 + erf ( b m +
i .sigma. u , i .times. 2 ) ) ( 1 - erf ( AL - i = 1 m .times.
.times. M w , i .times. ( b i + b m + i ) 2 .times. i = 1 m .times.
M w , i 2 .function. ( SISA i 2 + .sigma. u , i 2 ) ) )
##EQU40##
[0052] For the faulty case, satellite j.epsilon.{1, . . . , m} is
faulty, we get .sigma. L , i = .sigma. R , i = .sigma. i = SISA i
.times. .times. for .times. .times. all .times. .times. i = 1 ,
.times. , m .times. .times. with .times. .times. i .noteq. j and
.times. .times. .sigma. L , i = .sigma. R , i = .sigma. i = .sigma.
u , i - m .times. .times. for .times. .times. all .times. .times. i
= m + 1 , .times. , n .times. .times. and .times. .times. .sigma. L
, j = .sigma. R , j = .sigma. j = SISMA j ( 0.65 ) - b L , i = b R
, i = b i > 0 .times. .times. for .times. .times. all .times.
.times. i = 1 , .times. , n .times. .times. .times. with .times.
.times. i .noteq. j and .times. - b L , j = b R , j = b j .times. +
TH j ( 0.65 ' ) K L , i = K R , i = K i = 2 ( 1 + erf .function. (
b i .sigma. i .times. 2 ) ) - 1 for .times. .times. all .times.
.times. i = 1 , .times. , n , i . e . .times. inclusive .times.
.times. j ( 0.66 ) p w , int .function. ( AL ) .ltoreq. 2 n i = 1 n
.times. .times. ( 1 + erf .function. ( b i .sigma. i .times. 2 ) )
( 1 - erf .function. ( AL - M w , j .times. TH j - i = 1 n .times.
M w , i .times. b i 2 .times. i = 1 n .times. ( M w , i .times.
.sigma. i .times. ) 2 ) ) ( 0.67 ) ##EQU41##
[0053] As shown before several times (e.g. see inequalities (0.44)
and (0.45)) this formular can be written using the original
symbols. This will be omitted here.
[0054] More worthy of mention is the fact that different choices of
b.sub.i lead to different values for P.sub.w,int (AL). The optimal
choice, that gives smallest P.sub.w,int (AL), depends on the actual
satellite/user geometry. Therefore the user has to determine the
optimal choice of b.sub.i by himself.
Process for Mapping of ICAO Overbounding to Poem
[0055] As a third embodiment the process for mapping of ICAO
overbounding to POEM is described.
[0056] As already stated ICAO overbounding implies Galileo
overbounding. Therefore the method of mapping Galileo overbounding
to POEM described before is also applicable for ICAO
overbounding.
[0057] The question arising is whether it is possible to get a
factor smaller than 2 as necessary when mapping Galileo
overbounding to POEM by definition of q.sub.L/R.ident.2q for
mapping without bias and q'.sub.L(x).ident..theta.(-x)2q(x) and
q'.sub.R(x).ident..theta.(x)2q(x) for mapping with bias.
[0058] A positive function q.sub.L is part of POEM if inequality
.intg. - .infin. y .times. q L .function. ( x ) .times. d x
.gtoreq. .intg. - .infin. y .times. p .function. ( x ) .times. d x
##EQU42## is fulfilled for all y. For negative y we have the same
inequality for q itself. But for positive values of y we know
nothing but .intg. - .infin. y .times. p .function. ( x ) .times. d
x .ltoreq. 1. ##EQU43## Therefore the only condition we can use is
.intg. - .infin. y .times. q L .function. ( x ) .times. d x
.gtoreq. 1 ##EQU44## for all positive values of y. For reasons of
continuity the same is valid for y=0. Using the Gaussian type of
q.sub.L we calculate .intg. - .infin. y .times. q L .function. ( x
) .times. d x = K L 2 ( 1 + erf .function. ( y .sigma. L .times. 2
) ) . ##EQU45## Inserting y=0 leads to 1 .ltoreq. .intg. - .infin.
0 .times. q L .function. ( x ) .times. d x = K L 2 ( 1 + erf
.function. ( 0 .sigma. L .times. 2 ) ) = K L 2 K L .gtoreq. 2 ,
##EQU46## which answers the above question: we can not get a
smaller factor than 2 in the definition of q.sub.L.
[0059] A positive function q.sub.R is part of POEM if inequality 1
- K R + .intg. - .infin. y .times. q R .function. ( x ) .times. d x
.ltoreq. .intg. - .infin. y .times. p .function. ( x ) .times. d x
##EQU47## is fulfilled for all y. For positive y we have the same
inequality for q itself (because K.sub.R=1 for q.sub.R.ident.q).
But for negative values of y we know nothing but 0 .ltoreq. .intg.
- .infin. y .times. p .function. ( x ) .times. d x . ##EQU48##
Therefore the only condition that can be used is 1 - K R + .intg. -
.infin. y .times. q R .function. ( x ) .times. d x .ltoreq. 0
##EQU49## for all negative values of y. For reasons of continuity
the same is valid for y=0. Using now the Gaussian type of q.sub.R,
calculating the integral and inserting y=0 leads to 1 - .times. K
.times. R .times. 2 = .times. 1 - K .times. R + .times. K .times. R
.times. 2 ( 1 + erf .function. ( 0 .times. .sigma. R .times.
.times. 2 ) ) = .times. 1 - K .times. R + .intg. - .infin. .times.
0 .times. q .times. R .times. ( x ) .times. d x .ltoreq. 0 K
.times. R .gtoreq. 2 ##EQU50## which answers the above question: we
can not get a smaller factor than 2 in the definition of
q.sub.R.
[0060] The invention further discloses an apparatus that is
configured to execute the methods described above. Such an
apparatus could be part of one of the GNSS, e.g. a satellite of the
GNSS or some ground systems or the receiver of a user.
[0061] Although the invention has been described for GNSS and the
overbounding concepts Galileo Overbounding, ICAO Overbounding and
Paired Overbounding with Excess Mass. However, it is to be
understood by those skilled in the art that the invention is
neither limited to GNSS nor to Galileo Overbounding, ICAO
Overbounding and Paired Overbounding with Excess Mass,
respectively.
[0062] For further clarifications the following is disclosed:
[0063] Prerequisites: q .about. L := 2 .times. .chi. ( - .infin. ,
0 ] .times. q , q := 1 2 .times. .pi..sigma. .times. exp .function.
( - 1 2 .times. ( .sigma. ) 2 ) q L := K 2 .times. .pi. .times.
.sigma. .times. exp .function. ( - 1 2 .times. ( - .mu. .sigma. ) 2
) .times. .times. with .times. .times. .mu. < 0 ##EQU51##
[0064] Which conditions have to be stipulated on K and .mu., that
it holds:
[0065] Let z.gtoreq.0: It holds F q .about. L .ltoreq. F q L
.times. .times. F q .about. L .function. ( z ) = 1 .times. .ltoreq.
! .times. F q L .function. ( 0 ) .times. .ltoreq. monotonic .times.
.times. increasing .times. F q L .function. ( z ) = K 2 .times. ( 1
+ erf .function. ( z - .mu. 2 .times. .sigma. ) ) .times. 1 .times.
.ltoreq. ! .times. K 2 .times. ( 1 + erf ( - .mu. 2 .times. .sigma.
) ) . ( 1 ) ##EQU52## Then it holds F.sub.{tilde over
(q)}.sub.L(z).ltoreq.F.sub.q.sub.L(z) for z.gtoreq.0.
[0066] Set K := 2 ( 1 + erf ( - .mu. 2 .times. .sigma. ) ,
##EQU53## i.e. the smallest K will be used.
[0067] Let z<0: It is F q .about. L .function. ( z ) = .intg. -
.infin. z .times. 2 .times. x ( - .infin. , 0 ] .function. ( x )
.times. q .function. ( x ) .times. d x = 2 .times. .intg. - .infin.
z .times. q .function. ( x ) .times. .times. d x = 2 .times. F q
.function. ( z ) = 2 1 2 ( 1 + erf ( z 2 .times. .sigma. ) ) = 1 +
erf .function. ( z 2 .times. .sigma. ) , ##EQU54## i.e. (1) is
equivalent to 1 + erf .function. ( z 2 .times. .sigma. ) .times.
.ltoreq. ! .times. K 2 .times. ( 1 + erf .function. ( z - .mu. 2
.times. .sigma. ) ) . ##EQU55##
[0068] With the K from above this will be: 1 + erf .function. ( z 2
.times. .sigma. ) .times. .ltoreq. ! .times. 1 + erf .function. ( z
- .mu. 2 .times. .sigma. ) 1 + erf .function. ( - .mu. 2 .times.
.sigma. ) .gtoreq. 0 , .times. for .times. .times. .mu. < 0 .
##EQU56##
[0069] Abbreviate: {tilde over (z)}:=z/( {square root over
(2)}.sigma.), {tilde over (.mu.)}:=.mu./( {square root over
(2)}.sigma.) and multiply with (the positive) denominator of the
right side, then (1) is equivalent to ( 1 + erf .function. ( z
.about. ) ) .times. ( 1 + erf .function. ( - .mu. .about. ) )
.times. .ltoreq. ! .times. 1 + erf .function. ( z .about. - .mu.
.about. ) .times. .times. - 1 .revreaction. erf .function. ( z
.about. ) + erf .function. ( - .mu. .about. = .mu. .about. ) + erf
.function. ( z .about. ) erf .function. ( - .mu. .about. ) .times.
.ltoreq. ! .times. erf .function. ( z .about. - .mu. .about. )
.times. .times. .pi. 2 . .revreaction. f .function. ( t ) := exp
.function. ( - t 2 ) .times. .intg. 0 z .about. .times. f + .times.
.intg. 0 .mu. .about. .times. f + erf ( z .about. ) .times. .intg.
0 .mu. .about. .times. f .times. .ltoreq. ! .times. .intg. 0 z
.about. - .mu. .about. .times. f ( 2 ) ##EQU57##
[0070] Distinction of cases: 1. {tilde over (.mu.)}.ltoreq.{tilde
over (z)}<0 and 2. {tilde over (z)}<{tilde over (.mu.)}. So
let {tilde over (.mu.)}.ltoreq.{tilde over (z)}<0. Consider
because of erf .function. ( z .about. ) < 0 .times. .intg. 0
.mu. .about. .times. f > 0 < 0 : .intg. 0 z .about. .times. f
+ .times. .intg. 0 .mu. .about. .times. f .times. = z .about. <
0 , f .times. .times. symmetric .times. .times. at .times. .times.
0 .times. .intg. z .about. 0 .times. f + .intg. 0 .mu. .about.
.times. f = .intg. z .about. .mu. .about. .times. f . ##EQU58##
Notice |{tilde over (.mu.)}|.gtoreq.|{tilde over (z)}|.
[0071] Furthermore, because of {tilde over (.mu.)}.ltoreq.{tilde
over (z)}:{tilde over (z)}-{tilde over (.mu.)}=|{tilde over
(z)}-{tilde over (.mu.)}|, therefore .intg. 0 z .about. - .mu.
.about. .times. f = .intg. 0 z .about. - .mu. .about. .times. f .
##EQU59##
[0072] Now it holds |{tilde over (.mu.)}-|{tilde over
(z)}|=.parallel.{tilde over (.mu.)}|-|{tilde over
(z)}.parallel..ltoreq.|{tilde over (z)}-{tilde over (.mu.)}|, and f
is monotone decreasing for positive arguments, therefore it holds
.intg. z .about. .mu. .about. .times. f .ltoreq. .intg. 0 z .about.
- .mu. .about. .times. f : ##EQU60##
[0073] Altogether it has been shown: .intg. 0 z ~ .times. f +
.intg. 0 .mu. ~ .times. f + erf .times. .times. ( z ~ ) .intg. 0
.mu. ~ .times. f < .intg. 0 z ~ .times. f + .intg. 0 .mu. ~
.times. f = .times. .intg. z ~ .mu. ~ .times. f .ltoreq. .intg. 0 z
~ - .mu. ~ .times. f = .intg. 0 z ~ - .mu. ~ .times. f ,
##EQU61##
[0074] Now let {tilde over (z)}<{tilde over (.mu.)}, i.e. {tilde
over (z)}-{tilde over (.mu.)}<0.
[0075] As above (*), it follows .intg. 0 z ~ .times. f + .intg. 0
.mu. ~ .times. f = .intg. z ~ .mu. ~ .times. f = - .intg. .mu. ~ z
~ .times. f . ##EQU62##
[0076] Analogue it holds because of the symmetry of f for the right
side of (2): .intg. 0 z ~ - .mu. ~ .times. f = - .intg. z ~ - .mu.
~ 0 .times. f = - .intg. 0 - ( z ~ - .mu. ~ ) .times. f = - .intg.
0 .mu. ~ - z ~ .times. f . ##EQU63##
[0077] Therefore, (2) is equivalent to: - .intg. .mu. ~ z ~ .times.
f + erf .function. ( z ~ ) .times. .intg. 0 .mu. ~ .times. f
.times. .ltoreq. ! .times. - .intg. 0 .mu. ~ - z ~ .times. f .
.times. .times. .revreaction. erf .times. .times. ( z ~ ) = - erf
.times. .times. ( z ~ ) .times. .intg. 0 .mu. ~ - z ~ .times. f
.ltoreq. .intg. .mu. ~ z ~ .times. f + erf .times. .times. ( z ~ )
.times. .intg. 0 .mu. ~ .times. f . ( 3 ) ##EQU64##
[0078] Now showing (3): .intg. .mu. ~ z ~ .times. f + erf .times.
.times. ( z ~ ) .intg. 0 .mu. ~ .times. f = .times. .intg. .mu. ~ z
~ .times. f + ( 1 - erfc .times. .times. ( z ~ ) ) .intg. 0 .mu. ~
.times. f = .times. .intg. 0 z ~ .times. f - erfc .times. .times. (
z ~ ) .intg. 0 .mu. ~ .times. f = = .mu. ~ - z ~ < z ~ , for
.times. .times. .mu. ~ < 0 .times. .times. .intg. 0 .mu. ~ - z ~
.times. f + .intg. .mu. ~ - z ~ z ~ .times. f - erfc .function. ( z
~ ) > 0 .times. .intg. 0 .mu. ~ .times. f .gtoreq. .gtoreq. 0
.ltoreq. f .ltoreq. 1 .times. .times. .intg. 0 .mu. ~ - z ~ .times.
f + min x .di-elect cons. [ .mu. ~ - z ~ , z ~ ] .times. f
.function. ( x ) ( z ~ - ( .mu. ~ - z ~ ) ) = - z ~ - .mu. ~ + z ~
= .mu. ~ - erfc .function. ( z ~ ) .times. max x .di-elect cons. [
0 , .mu. ~ ] .times. f .times. ( x ) = 1 ( .mu. ~ - 0 ) = f .times.
.times. is .times. .times. monotone .times. .times. decreasing
.times. for .times. .times. positive .times. .times. arguments
.times. .times. .intg. 0 .mu. ~ - z ~ .times. f + f .function. ( z
~ ) .mu. ~ - erfc .times. .times. ( z ~ ) .mu. ~ ( 4 )
##EQU65##
[0079] Now it holds erfc .times. .times. ( x ) .ltoreq. f
.function. ( x ) .times. .times. for .times. .times. x > 0.
##EQU66## erfc .times. .times. ( x ) = .times. 2 .pi. .times.
.intg. x .infin. .times. exp .function. ( - t 2 ) .times. d t
.times. = `` .times. du = dt '' u := t - x .times. 2 .pi. .times.
.intg. 0 .infin. .times. exp .function. ( - ( u + x ) 2 ) .times. d
u = .times. 2 .pi. .times. .intg. 0 .infin. .times. exp .function.
( - u 2 ) exp .function. ( - 2 .times. .times. ux - x 2 ) .times. d
u = .times. exp .function. ( - x 2 ) .times. 2 .pi. .times. .intg.
0 .infin. .times. exp .function. ( - u 2 ) > 0 .times. exp
.function. ( - 2 .times. .times. ux ) 0 < .times. .ltoreq. 1 ,
for .times. .times. x > 0 , u .gtoreq. 0 .times. d u .ltoreq.
.times. exp .function. ( - x 2 ) .times. 2 .pi. .times. .intg. 0
.infin. .times. exp .function. ( - u 2 ) .times. d u = .times. exp
.function. ( - x 2 ) .ident. f .function. ( x ) ##EQU66.2##
[0080] Therefore, it holds because of (4) .intg. .mu. ~ z ~ .times.
f + erf .function. ( z ~ ) .intg. 0 .mu. ~ .times. f .gtoreq.
.intg. 0 .mu. ~ - z ~ .times. f + f .function. ( z ~ ) .mu. ~ - f
.function. ( z ~ ) .mu. ~ = .intg. 0 .mu. ~ - z ~ .times. f ,
.times. that .times. .times. is .times. .times. ( 3 ) .times.
.times. for .times. .times. z < .mu. . ##EQU67##
[0081] Because (3)(2)(1), everything has been shown.
[0082] Add on for faulty case: Satellite j faulty! Shift
TH.sub.j.
[0083] Overbounding condition without shift by TH.sub.j fulfilled,
i.e. q := 1 2 .times. .pi. .times. .sigma. .times. exp .function. (
- 1 2 .times. ( . .sigma. ) 2 ) ##EQU68## q ~ L , S := 2 .times.
.chi. ( - .infin. , - TH ] .times. q .function. ( + TH ) , .times.
q ~ R , S := 2 .times. .chi. ( 0 , .infin. ) .times. q .function. (
- TH ) , .times. q L , S := K 2 .times. .pi. .times. .sigma.
.times. exp .function. ( - 1 2 .times. ( + TH - .mu. L .sigma. ) 2
) .times. .times. with .times. .times. .mu. L < 0 ##EQU69## q R
, S := K 2 .times. .pi. .times. .sigma. .times. exp .function. ( -
1 2 .times. ( - TH - .mu. R .sigma. ) 2 ) .times. .times. with
.times. .times. .mu. R > 0 ##EQU69.2## F q ~ L , S .ltoreq. F q
L , S .times. : .times. F q ~ L , S .function. ( z ) = F q ~ L
.function. ( z + TH ) .ltoreq. F q L .function. ( z + TH ) = F q L
, S .function. ( z ) ##EQU69.3## (z) with K.sub.L as above!
[0084] (Let z.gtoreq.-TH: It holds ( Let .times. .times. z .gtoreq.
- TH .times. : .times. .times. It .times. .times. holds .times.
.times. F q ~ L , S .function. ( z ) = 1 .times. .ltoreq. ! .times.
F q L , S .function. ( - TH ) .ltoreq. F q L , S .function. ( z ) =
F q L .function. ( z + TH ) = K 2 .times. ( 1 + erf .function. ( z
- ( .mu. L - TH ) 2 .times. .sigma. ) ) .times. .times. ok ) 1
.ltoreq. K 2 .times. ( 1 + erf .function. ( - .mu. L 2 .times.
.sigma. ) ) ##EQU70##
[0085] Claim: F q ~ R , S * .ltoreq. F p .function. ( - TH ) , F p
.function. ( + TH ) .ltoreq. F q ~ L , S .times. : .times. .times.
F q ~ L , S .function. ( z ) = 2 .times. .intg. - .infin. z .times.
.chi. ( - .infin. , - TH ] .function. ( x ) .times. q .function. (
x + TH ) .times. d x .times. = y := x + TH .times. 2 .times. .intg.
- .infin. z + TH .times. .chi. ( - .infin. , - TH ] .function. ( y
- TH ) = .chi. ( - .infin. , 0 ] .function. ( y ) .times. q
.function. ( y ) .times. d y .times. .times. = .intg. - .infin. z +
TH .times. 2 .times. .times. .chi. ( - .infin. , 0 ] .times. 2
.times. .times. .chi. ( - .infin. , 0 ] .function. ( y ) .times. q
.function. ( y ) .times. d y = F q ~ L .function. ( z + TH )
##EQU71##
[0086] Therefore, it holds F p .function. ( + TH ) .function. ( z )
= .intg. - .infin. z .times. p .function. ( x + TH ) .times. d x =
.intg. - .infin. z + TH .times. p .function. ( y ) .times. d y = F
p .function. ( z + TH ) .ltoreq. F q ~ L .function. ( z + TH )
.ltoreq. F q ~ L , S .function. ( z ) .times. F q ~ R , S *
.function. ( z ) = 1 - K R = 1 + F q ~ R , S .function. ( z )
.times. = as .times. .times. above .times. F q ~ R .function. ( z -
TH ) .ltoreq. F p .function. ( z - TH ) = F p .function. ( - TH )
.function. ( z ) .times. .times. ok .times. ##EQU72##
[0087] The foregoing disclosure has been set forth merely to
illustrate the invention and is not intended to be limiting. Since
modifications of the disclosed embodiments incorporating the spirit
and substance of the invention may occur to persons skilled in the
art, the invention should be construed to include everything within
the scope of the appended claims and equivalents thereof.
* * * * *